Limit Cycles: Existence and Non-existence Criteria

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Taught by OCW
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Lesson Summary:

In this lesson, we learn about limit cycles, which are closed trajectories in a nonlinear autonomous system that represent periodic behavior. Limit cycles are isolated and stable, meaning nearby points approach the cycle, and they are of interest in modeling natural phenomena that exhibit periodic motion. The existence of limit cycles is difficult to determine, but two theorems are presented for non-existence criteria: Bendixson's criterion and a criterion involving critical points. Bendixson's criterion involves calculating the divergence of the vector field and can be proven using 1802-level math.

Lesson Description:

Limit Cycles: Existence and Non-existence Criteria -- Lecture 32. Learn about limit cycles and their criteria.

Arthur Mattuck, 18.03 Differential Equations, Spring 2006. (Massachusetts Institute of Technology: MIT OpenCourseWare), (Accessed November 29, 2008). License: Creative Commons BY-NC-SA.
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Questions answered by this video:
  • What are limit cycles?
  • What are the criteria for limit cycles to exist?
  • What is a closed trajectory?
  • What does it mean for a limit cycle to be isolated?
  • What does it mean for a limit cycle to be stable?
  • How do you know when a limit cycle will not exist?
  • What is Poincare-Bendixson criteria?
  • Staff Review

    • Currently 4.0/5 Stars.
    This lecture starts off with a bit of review from the last few lectures, including trajectories and velocity fields. This lecture is devoted entirely to the idea of limit cycles and how to determine whether a limit cycle exists. There are some example problems done as well. A very interesting topic explained very well.